Scalar-tensor theories have been studied extensively as a model of inflation/dark energy, and there has been a growing interest in incorporating higher-derivative interactions of the scalar field into the action without introducing Ostrogradsky ghost [1]}, [2]}.
It was shown that the ghost can be
eliminated if the higher-derivative terms are degenerate [3]}, [4]}, [5]}, [6]}, [7]}, [8]}.
With the knowledge of the degeneracy conditions, one can systematically construct ghost-free scalar-tensor theories, which are known as degenerate higher-order scalar-tensor (DHOST) theories [4]}, [10]}, [11]}, [12]}, [13]}.Relaxing the degeneracy conditions so that the higher-derivative terms are degenerate only in the unitary gauge, we obtain a broader class of scalar-tensor theories [14]}, [15]}, [16]}, [17]}, [18]}.
In that case, there is an apparent extra DOF in a generic gauge, but it satisfies an elliptic differential equation and hence is an instantaneous (or “shadowy”) mode [15]}, [18]}.
The DHOST theory includes the Horndeski [21]}, [22]}, [23]} and the Gleyzes-Langlois-Piazza-Vernizzi theories [24]} as special cases, and hence form a general class of healthy scalar-tensor theories (see [25]}, [26]} for reviews).
